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Differentially positive systemsF. Forni, R. Sepulchre

Abstract—The paper introduces and studies differentially pos-itive systems, that is, systems whose linearization along an arbi-trary trajectory is positive. A generalization of Perron Frobeniustheory is developed in this differential framework to show thatthe property induces a (conal) order that strongly constrainsthe asymptotic behavior of solutions. The results illustrate thatbehaviors constrained by local order properties extend beyondthe well-studied class of linear positive systems and monotonesystems, which both require a constant cone field and a linearstate space.

I. INTRODUCTION

Positive systems are linear behaviors that leave a coneinvariant [11]. They have a rich history both because of therelevance of the property in applications (e.g., when modelinga behavior with positive variables [30], [18], [25]) and becausethe property significantly restricts the behavior, as establishedby Perron Frobenius theory: if the cone invariance is strict,that is, if the boundary of the cone is eventually mappedto the interior of the cone, then the asymptotic behavior ofthe system lies on a one dimensional object. Positive systemsfind many applications in systems and control, ranging fromspecific stabilization properties [48], [32], [18], [14], [26], [39]to observer design [22], [9], and to distributed control [31],[37], [42].

Motivated by the importance of positivity in linear systemstheory, the present paper investigates the behavior of differ-entially positive systems, that is, systems whose linearizationalong trajectories is positive. We discuss both the relevanceof the property for applications and how much the propertyrestricts the behavior, by generalizing Perron Frobenius theoryto the differential framework. The conceptual picture is thata cone is attached to every point of the state space, defininga cone field, and that contraction of that cone field along theflow eventually constrains the behavior to be one-dimensional.

Differential positivity reduces to the well-studied propertyof monotonicity when the state-space is a linear vector spaceand when the cone field is constant. First studied for closedsystems [43], [24], [23], [13] and later extended to opensystems [3], [5], [4], the concept of monotone systems en-compasses cooperative and competitive systems [25], [36] andis extensively adopted in biology and chemistry for modelingand control purposes [15], [17], [16], [45], [6], [7]. Differential

F. Forni and R. Sepulchre are with the University of Cambridge,Department of Engineering, Trumpington Street, Cambridge CB21PZ, and with the Department of Electrical Engineering andComputer Science, University of Liege, 4000 Liege, Belgium,ff286@cam.ac.uk|r.sepulchre@eng.cam.ac.uk. The researchis supported by FNRS. The paper presents research results of the BelgianNetwork DYSCO (Dynamical Systems, Control, and Optimization), fundedby the Interuniversity Attraction Poles Programme, initiated by the BelgianState, Science Policy Office. The scientific responsibility rests with itsauthors.

positivity is an infinitesimal characterization of monotonic-ity. The differential viewpoint allows for a generalization ofmonotonicity because the state-space needs not be linear andthe cone needs not be constant in space. The generalizationis relevant in a number of applications. In particular, non-constant cone fields in linear spaces and invariant cone fieldson nonlinear spaces are two situations frequently encounteredin applications. Like monotonicity, differential positivity in-duces an order between solutions. But in contrast to monotonesystems, the conal order needs not to induce a partial orderglobally, allowing for instance to (locally) order solutions onclosed curves, such as along limit cycles or in nonlinear spacessuch as the circle.

A main contribution of the paper is to generalize Perron-Frobenius theory in the differential framework. The Perron-Frobenius vector of linear positive systems here becomes avector field and the integral curves of the Perron-Frobeniusvector field shape the attractors of the system. A main resultof the paper is to provide a characterization of limit setsof differentially positive systems akin to Poincare-Bendixsontheorem for planar systems. Differentially positive systems canmodel multistable behaviors, excitable behaviors, oscillatorybehaviors, but preclude for instance the existence of attractivehomoclinic orbits, and a fortiori of strange attractors. In thatsense, differentially positive systems single out a significantclass of nonlinear systems that have a simple asymptoticbehavior.

The paper is organized as follows. Section II introduces themain ideas of differential positivity on familiar phase portraitsand at an intuitive level. It aims at showing that the differentialconcept of positivity is a natural one. Section III coverssome mathematical preliminaries and notations while SectionIV summarizes the main mathematical notions of order onmanifolds. The next three sections contain the main results ofthe paper: the formal notion of differentially positive system,differential Perron-Frobenius theory, and a characterizationof limit sets of differentially positive systems. Section VIIIillustrates several important points of the paper on the popularnonlinear pendulum example. Proofs are in appendix. Ourtreatment of differential positivity is for continuous-time anddiscrete-time open systems. The important topic of intercon-nections of differentially positive systems is a rich one andwill be discussed in a separate paper.

II. DIFFERENTIAL POSITIVITY IN A NUTSHELL

Figure 1 illustrates four different phase portraits of (closed)differentially positive systems. Two of the phase portraits arerepresented in two different set of coordinates. The figureillustrates that for each of the phase portraits, a cone canbe attached at any point in such a way that the cone is

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Figure 1. The phase portraits of four different planar differentially positivesystems. (I) a linear consensus model. (II) a monotone bistable model. (III)the harmonic oscillator. (IV) the nonlinear pendulum with strong dampingand torque.

infinitesimally contracted by the flow (i.e. the cone angleshrinks under the action of the flow). Furthermore, the conescan be patched to each other to define a smooth cone field.

In the first two examples, the cone is actually the sameeverywhere, defining a constant cone field in a linear space.In the third example, both the state-space and the dynamicsare linear but the cone rotates with the flow. It defines a nonconstant cone field in a linear space. In the fourth example,the cone field must be defined infinitesimally because the statespace is nonlinear. At each point, a cone is defined in thetangent space. The nonlinear cylindrical space S × R is aLie group and the cone is moved from point to point by(left) translation. The analogy between the third and fourthexamples is apparent when studying the phase portrait ofthe harmonic oscillator in polar coordinates. The nonlinearchange of coordinates makes the cone invariant on the conicnonlinear space R+×S. The analogy between the first, second,and fourth examples is apparent when unwrapping the phaseportrait of the nonlinear pendulum in the plane. In cartesiancoordinates, that is, unwrapping the angular coordinate ϑ onthe real line, the cone field becomes constant in a linear space.

The first phase portrait is the phase portrait of a linearsystem that leaves the positive orthant invariant. It is a strictlypositive system. Its behavior is representative of consensusbehaviors extensively studied in the recent years [31], [35],[41]. The second phase portrait leaves the same cone invariantbut the dynamics are nonlinear. Here the cone invariance

can be characterized differentially: the linearization along anytrajectory is a positive linear system with respect to the positiveorthant. It is an example of monotone system, representativeof bistable behaviors extensively studied in decision-makingprocesses, see e.g. [47]. The third example is the phaseportrait of the harmonic oscillator. Solutions cannot be globallyordered in the state space because the trajectories are closedcurves. But the positivity of the linearization is neverthelessapparent in polar coordinates. The corresponding order prop-erty will be characterized by the notion of conal order onmanifolds developed in Section IV. The fourth example is thephase portrait of the nonlinear pendulum with strong damping.Positivity of the linearization and differential positivity of thenonlinear pendulum is studied in details in the last Section ofthe paper.

The main message of the paper is that differential positivityconstrains the asymptotic behavior of the four different phaseportraits in a similar way. For linear positive systems, thisis Perron- Frobenius theory. The Perron-Frobenius vector at-tracts all solutions to a one-dimensional ray. For differentiallypositive systems, the generalized object is a Perron-Frobeniuscurve, an integral curve of the Perron-Frobenius vector fieldcharacterized in Section V. In the second phase portrait, this isthe heteroclinic orbit connecting the two stable equilibria andthe unstable saddle equilibrium. In the third phase portrait,every trajectory is a Perron-Frobenius curve. The differentialpositivity is not strict in that case. In the fourth phase portrait,all solutions except the unstable equilibrium are attracted to asingle Perron-Frobenius curve, the limit cycle.

The convergence properties of differentially positive sys-tems are a consequence of the infinitesimal contraction ofcones along trajectories. The significance of the property isthat it can be checked locally but that it discriminates amongdifferent types of global behaviors. The smoothness of thecone field is what connects the local property to the globalproperty. A most important feature of differential positivityis that it allows saddle points such as in Figure 1.II, becausethe local order is compatible with a global smooth cone field,but that it does not allow saddle points such as in Figure 2.The homoclinic orbit makes the local order dictated by thesaddle point incompatible with a global smooth cone field.This incompatibility has been recognized since the early workof Poincare as the essence of complex behaviors. In contrast,the limit sets of differentially positive systems are simple, ina sense that is made precise in Section VII.

?

Figure 2. Strict differential positivity excludes homoclinic orbits on saddlepoints whose unstable manifold has dimension one. The local order imposedby the saddle point cannot be extended globally to a smooth cone field.

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III. MATHEMATICAL PRELIMINARIESAND BASIC ASSUMPTIONS

A (d-dimensional) manifold X is a couple (X ,A+) whereX is a set and A+ is a maximal atlas of X into Rd, suchthat the topology induced by A+ is Hausdorff and second-countable (we refer to this topology as the manifold topology).Throughout the paper every manifold is connected. TxXdenotes the tangent space at x and TX :=

⋃x∈X {x} × TxX

denotes the tangent bundle. X is endowed with a Riemannianmetric tensor, represented by a (smoothly varying) innerproduct 〈·, ·〉x . |δx|x :=

√〈δx, δx〉x, for any δx ∈ TxX . The

Riemannian metric endows the manifold with the Riemanniandistance D. We assume that (X , D) is a complete metric space(see e.g. [1, Section 3.6]). The metric space topology and themanifold topology agree [10, Theorem 3.1].

Given two smooth manifolds X1, X2, the differential off : X1 → X2 at x is denoted by ∂f(x) : TxX →Tf(x)X . Given X1 = Z × Y , the operator ∂zf(z, y) satisfies∂zf(z, y)δz := ∂f(z, y)[δz, 0] for each (z, y) ∈ X1 and each[δz, 0] ∈ T(z,y)X1, where δz ∈ TzZ . Finally, we write f(·, y)to denote the function in Z → X2 mapping each z ∈ Z intof(z, y) ∈ X2.

A curve or γ on X , is a mapping γ : I → X where eitherI ⊆ R or I ⊆ Z. domγ and imγ denote domain and imageof γ. We say that a curve γ : I → X is bounded if imγis a bounded set. We sometime use γ(s) or dγ(s)

ds to denote∂γ(s)1, for s ∈ domγ.

Given a set S ⊆ X , intS and bdS denote interior andboundary of S, respectively. Given a vector space V , a set S ⊆V , and a constant λ ∈ R, λS denotes the set {λx ∈ V |x ∈ S}.S + S denotes the set {x+ y ∈ V |x, y ∈ S}. Given a pointy ∈ S , S\{y} := {x ∈ S |x 6= p}. Given a sequence of setsSn, limn→∞ Sn is the usual set-theoretic limit based on thePainleve-Kuratowski convergence [38, Chapter 4].

Let Σ be an open continuous dynamical system with(smooth) state manifold X and input manifold U , representedby x = f(x, u), (x, u) ∈ X × U , where f is a (input-dependent) vector field that assigns to each (x, u) ∈ X × Ua tangent vector f(x, u) ∈ TxX . We make the standingassumptions that the vector field f and u(·) are C2 functions.Following [10, Chapter 4, Section 4], two differentiable curvesx(·) : I ⊆ R → X (trajectory) and u(·) : I ⊆ R → U (input)are a solution pair (x(·), u(·)) ∈ Σ if x(t) = f(x(t), u(t)) forall t ∈ I . An open discrete dynamical system Σ is representedby the recursive equation x+ = f(x, u), (x, u) ∈ X × U . Wemake the standing assumption that f : X × U → X is a C1

function. (x(·), u(·)) : [t0,∞) ⊆ Z → X × U is a solutionpair of Σ if x(·) and u(·) satisfy x(t+ 1) = f(x(t), u(t)) foreach t ∈ [t0,∞).

In what follows, we make the simplifying assumption offorward completeness of the solution space, namely that everysolution pair has domain I = [t0,∞) ⊆ R (⊆ Z). Given thesolution pair (x(·), u(·)) : [t0,∞)→ X × U ∈ Σ we say thatψ(·, t, x(t), u(·)) := x(·) is the trajectory or the integral curvepassing through x(t) at time t ≥ t0 under the action of theinput u(·). For constant inputs we simply write ψ(·, t, x(t), u)and for closed systems we use ψ(·, t, x(t)). The flow of Σ is

given by the quantity ψ(t, t0, ·, u) for any t ≥ t0. For any curveγ(·) and set S, ψ(t, t0, γ(·), u) denotes the time evolution ofγ(·) along the flow of the system at time t, and ψ(t, t0,S, u)denotes the set {ψ(t, t0, x, u) |x ∈ S}. For closed systems wesay that xω ∈ X is an ω-limit point of a trajectory x(·) ifthere exists a sequence of times tk →∞ as k →∞ such thatxω = limk→∞ x(tk). In a similar way, an α-limit point of atrajectory x(·) is given by limk→∞ x(tk) for some sequencetk → −∞ as k → ∞. The ω-limit set ω(x0) (α-limit set) isthe union of the ω-limit points (α-limit points) of the trajectoryx(·) from the initial condition x(t0) = x0.

IV. CONE FIELDS, CONAL CURVES, CONAL ORDERS

A conal manifold X is a smooth manifold endowed with acone field [28],

KX (x) ⊆ TxX ∀x ∈ X . (1)

Like for vector fields, a cone field attaches to each point x ofthe manifold a cone KX (x) defined in the tangent space TxX .Throughout the paper, each cone KX (x) ⊆ TxX is closed,pointed and convex (for each x ∈ X , KX (x) + KX (x) ⊆KX (x), λKX (x) ⊆ KX (x) for any λ ∈ R≥0, and KX (x) ∩−KX (x) = {0}). To avoid pathological cases, we assumethat each cone is solid (i.e. it contains n independent tangentvectors, where n is the dimension of the tangent space) andthere exists a linear invertible mapping Γ(x1, x2) : Tx1

X →Tx2X for each x1, x2 ∈ X , such that Γ(x1, x2)KX (x1) =

KX (x2).Note that the application of a linear invertible mapping to

a cone is intended as an operation on the rays of the cone,that is, Γ(x1, x2)KX (x1) := {λΓ(x1, x2)δx ∈ Tx2X | δx ∈KX (x1), λ ≥ 0}.

We make the standing assumption that each cone field issmooth. In particular, in local coordinates,

KX (x) = {δx ∈ TxX | ∀i ∈ I, ki(x, δx) ≥ 0} (2)

where I ⊆ Z is an index set and ki : TX → R are functions;and we say that a cone field is smooth if the functions ki aresmooth.

A curve γ : I ⊆ R→ X is a conal curve on X if

γ(s) ∈ KX (γ(s)) for all s ∈ I . (3)

Conal curves are integral curves of the cone field, as shown inFigure 3. They endow the manifold with a local partial order:for each x1, x2 ∈ X , x1 vKX x2 if and only if there existsa conal curve γ : I ⊆ R → X such that γ(s1) = x1 andγ(s2) = x2 for some s1 ≤ s2.

The conal order vKX is the natural generalization onmanifolds of the notion of partial order on vector spaces.In fact vKX is a partial order when X is a vector space

γ(s)

KX (γ(s))γ(·)

Figure 3. A conal curve satisfies (3). If s1 ≤ s2 then γ(s1) vKX γ(s2).

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and the cone field KX (x) = KX is constant: two pointsx, y ∈ X satisfy x vKX y iff y − x ∈ KX , as shown in [28,Proposition 1.10], which is the usual definition of a partialorder on vector spaces [40, Chapter 5]. In general, vKX isnot a (global) partial order on X since antisymmetry may fail.The reader is referred to [28] and [33] for a detailed expositionof the relations among cone fields, ordered manifolds, andhomogeneous spaces.

Example 1: For the manifold S×R in Figure 1.IV, the conalorder given by the cone field δθ ≥ 0, δθ + δv ≥ 0 is not apartial order since, for any pair of points x, y ∈ S× R, thereexists a conal curve connecting x to y and viceversa. However,in a sufficiently small neighborhood of any point x, the conalorder is a partial order. y

V. DIFFERENTIALLY POSITIVE SYSTEMS

A. Definitions

A dynamical system is differentially positive when itslinearization is positive. Positivity is intended here in the senseof cone invariance [11]. More precisely, a dynamical systemΣ on the conal state-input manifold X × U is differentiallypositive when the cone field

K(x, u) = KX (x, u)︸ ︷︷ ︸⊆TxX

×KU (x, u)︸ ︷︷ ︸⊆TuU

⊆ T(x,u)X × U (4)

is invariant along the trajectories of the linearized system. Fordiscrete-time system x+ = f(x, u), the invariance propertyhas a simple formulation. The mapping f : X × U → X , isdifferentially positive if, for all x ∈ X and all u, u+ ∈ U ,

∂f(x, u)K(x, u) ⊆ KX (f(x, u), u+) . (5)

Indeed, ∂f(x, u) is a positive linear operator, mapping eachtangent vector (δx, δu) ∈ K(x, u) ⊆ T(x,u)X × U intoδx+ := ∂f(x, u)[δx, δu] ∈ KX (x+, u+) ⊆ Tx+X . A graph-ical representation for closed discrete systems is provided inFigure 4. The relation between the positivity of the operator∂f(x, u) in (5) and the positivity of the linearization of Σ isjustified by the fact that δx+ = ∂xf(x, u)δx + ∂uf(x, u)δu,which establishes the positivity of the linearized dynamics inthe sense of [11], [18], [14], [2].

KX (x)x

x+

KX (x+)

∂f(x)KX (x)

Figure 4. A graphical representation of condition (5) for the (closed) discretetime system x+ = f(x). The cone field K(x, u) reduces to KX (x) in thiscase.

For general continuous-time x = f(x, u) ∈ TxX anddiscrete-time x+ = f(x, u) ∈ X dynamical systems Σ((x, u) ∈ X × U), the definition of differential positivityinvolves the prolonged system δΣ introduced in [12]

δΣ :

{(x+) x = f(x, u)

(δx+) ˙δx = ∂xf(x, u)δx+ ∂uf(x, u)δu .(6)

We call variational component the second equation of (6).Definition 1: Σ is a differentially positive dynamical system

(with respect to K in (4)) if, for all t0 ∈ R, any solution pair((x, δx)(·), (u, δu)(·)) : [t0,∞)→ TX ×TU ∈ δΣ leaves thecone field K invariant. Namely,

{δx(t0) ∈ KX (x(t0), u(t0))δu(t) ∈ KU (x(t), u(t)), ∀t ≥ t0

⇒δx(t) ∈ KX (x(t), u(t)), ∀t ≥ t0 .

(7)

y

In continuous-time, differential positivity of Σ is thuspositivity of the linearized system ˙δx = A(t)δx + B(t)δualong any solution pairs (x(·), u(·)) ∈ Σ, where A(t) :=∂xf(x(t), u(t)) and B(t) := ∂uf(x(t), u(t)). For closedsystems x = f(x), with cone field KX (x) ⊆ TxX , wehave ˙δx = A(t)δx, where A(t) := ∂f(x(t)), along anygiven solution x(·) : [t0,∞) → X ∈ Σ. Therefore, thefundamental solution Ψx(·)(t, t0) of the linearized dynamics[44, Appendix C.4] satisfies Ψx(·)(t, t0)KX (x) ⊆ KX (x) foreach t ∈ [t0,∞), that is, Ψx(·)(t, t0) is a positive linearoperator.

Strict differential positivity is to differential positivity whatstrict positivity is to positivity. We anticipate that this (mild)property will have a strong impact on the asymptotic behaviorof differentially positive systems, as shown in Section VI.

Definition 2: Σ is (uniformly) strictly differentially positive(with respect to K) if differential positivity holds and thereexists T > 0 and a cone field RX (x, u) ⊆ intKX (x, u) ∪{0} such that, for all t0 ∈ R, any ((x, δx)(·), (u, δu)(·)) :[t0,∞)→ TX × TU ∈ δΣ satisfies

{δx(t0) ∈ KX (x(t0), u(t0))δu(t) ∈ KU (x(t), u(t)), ∀t ≥ t0

⇒δx(t) ∈ RX (x(t), u(t)), ∀t ≥ t0 + T .

(8)

We assume that the cone field RX also satisfies the followingadditional technical condition:

Γ(x1, u1, x2, u2)RX (x1, u1) = RX (x2, u2) (9)

for each (x1, u1), (x2, u2) ∈ X ×U , where Γ(x1, u1, x2, u2) :Tx1X → Tx2X is a linear invertible mapping such thatΓ(x1, u1, x2, u2)KX (x1, u1) = KX (x2, u2) (see Section IV).

yFor open systems with output h : X × U → Y – Y output

manifold, endowed with the cone field KY(y) ∈ TyY – thenotion of (strict) differential positivity requires the furthercondition that h is a differentially positive mapping, that is,∂h(x, u)K(x, u) ⊆ KY(h(x, u)), for each (x, u) ∈ X × U .

Remark 1: Differential positivity has a geometric character-ization. restricting to closed systems for simplicity, considerthe cone field KX (x) represented by (2) where I is an index setand ki are smooth functions. Then, (7) is equivalent to requirethat ki(x(t), δx(t)) ≥ 0 along any solution (x(·), δx(·)) ∈ δΣ,for all i ∈ I . Therefore, differential positivity for a discretesystem can be established by testing that ∀i ∈ I, ki(x, δx) ≥ 0implies ∀i ∈ I, k+

i := ki(f(x), ∂f(x)δx) ≥ 0, for each(x, δx) ∈ TX . In a similar way, for continuous systems,consider any pair (x, δx) ∈ TX such that ki(x, δx) ≥ 0 for

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all i ∈ I and test that, for any j ∈ I , if kj(x, δx) = 0 thenkj := ∂kj(x, δx)[f(x), ∂f(x)δx] ≥ 0. y

B. Examples

1) Positive linear systems are differentially positive: Con-sider the dynamics Σ given by x+ = Ax on the vector space V .Positivity with respect to the cone KV ⊆ V reads AKV ⊆ KV ,[11]. A typical example is provided by the case of a matrix Awith non-negative entries which guarantees the invariance ofthe positive orthant KV := Rn+.

Since each tangent space of a vector space can be identifiedto the vector space itself, i.e. TxV = V for each x ∈ V ,consider the manifold X := V and define the lifting of thecone KV to the cone field KX (x) := KV ⊆ TxX , for eachx ∈ X (constant cone field). Then the linearized dynamicsreads δx+ = Aδx and the prolonged system trivially satisfiesAKX (x) ⊆ KX (Ax).

2) Monotone systems are differentially positive: A mono-tone dynamical system [43], [3] is a dynamical system whosetrajectories preserve some partial order relation on the statespace. Moving from closed [43], [24], [23], [13], [25], [36]to open systems [3], [5], [4], this wide class of systemsis extensively adopted in biology and chemistry both formodeling and control [15], [17], [16], [45], [6], [7].

The partial order � of a monotone system is typicallyinduced by a conic subset KV ⊆ V of the state (vector) spaceV . Precisely, two points x, x ∈ V satisfy x �KV x if andonly if x−x ∈ KV . The preservation of the order along thesystem dynamics reads as follows: if x(·), x(·) ∈ Σ satisfyx(t0) �KV x(t0) for some initial time t0, then x(t) �KV x(t)for all t ≥ t0, [43].

To show that a monotone system is differentially positive,consider V as a manifold endowed with the constant conefield KV(x) := KV , x ∈ V . By monotonicity, the infinitesimaldifference between two ordered neighboring solutions δx(·) :=x(·) − x(·) satisfies δx(t) ∈ KV(x(t)), for each t ≥ t0.Differential positivity follows from the fact that (x(·), δx(·))is a trajectory of the prolonged system δΣ.

Theorem 1: Given any cone KV on the vector space V , thepartial order �KV , and the cone field KX (x) := KV , a (closed)dynamical system is monotone if and only if is differentiallypositive. y

Proof: For constant cone fields on vector spaces recallthat �KV and vKV are equivalent relations (see Section IV).Consider a conal curve γ(t0, ·) connecting two ordered initialpoints γ(t0, 0) := x(t0) �KV x(t0) =: γ(t0, 1). Note thatγ(t0, s1) vKV γ(t0, s2) for each s1 ≤ s2. For each s ∈ [0, 1],let γ(·, s) be a trajectory of Σ. Indeed, γ(t, ·) representsthe time evolution of the curve γ(t0, ·) along the flow ofthe system. [⇐] Differential positivity guarantees that γ(t, ·)is a conal curve for each t ≥ t0. This follows from thefact that the pair (xs(t), δxs(t)) := (γ(t, s), ddsγ(t, s)) is atrajectory of the prolonged system δΣ for each s ∈ [0, 1].Thus, x(t) vKV x(t) for all t ≥ t0. [⇒] Monotonicityguarantees that γ(t, s1) �KV γ(t, s2) for all s1 ≤ s2. Bya limit argument, d

dsγ(t, s) ∈ KV(γ(t, s)) for all t ≥ t0and al s ∈ [0, 1]. Thus γ(t, ·) is a conal curve. Note that

(xs(t), δxs(t)) := (γ(t, s), ddsγ(t, s)) is a trajectory of theprolonged system. Since γ(t0, ·) is a generic conal curve, (7)follows.

A similar result holds for open monotone systems, whichare typically characterized by introducing two orders �KX and�KU , respectively induced by the cone KX on the state spaceX and KU on the input space U , [3, Definition II.1]. Extendingthe argument above it is possible to show that a dynamicalsystem Σ is monotone with respect to (�KX ,�KU ) if andonly if Σ is differentially positive on the vector space X ×Uendowed with the constant cone field K(x, u) := KX × KU ,for each (x, u) ∈ X × U . In this sense, differential positivityon vector spaces and constant cone fields is the differentialformulation of monotonicity.

3) Differential positivity of cooperative systems and theKamke condition: A cooperative system x = f(x) with statespace X := Rn is monotone with respect to the partialorder induced by the positive orthant Rn+, thus differentiallypositive with respect to the cone field KX (x) := Rn+, x ∈ X .Exploiting the geometric conditions of Remark 1, differentiallypositivity with respect to KX holds when

[∂f(x)]ij ≥ 0 for all 1 ≤ i 6= j ≤ n, x ∈ X , (10)

where [·]ij denotes the ij component. To see this, define Ei asthe vector whose i-th element is equal to one and the remainingto zero and note that the positive orthant is defined by the setof δx that satisfy 〈Ei, δx〉 ≥ 0. Then, from Remark 1, theinvariance reads 〈Ei, δx〉 = 0 ⇒ 〈Ei, ∂f(x)δx〉 ≥ 0 for anyx ∈ X , δx ∈ KX (x) = Rn+, and i ∈ {1, . . . , n}. (10) followsby selecting δx = Ej 6= Ei. Indeed, ∂xf(x) is a Metzlermatrix for each x ∈ X [3, Section VIII].

Cooperative systems typically satisfy (10), as shown in [43,Remark 1.1] on closed systems. A similar result is providedin [3, Proposition III.2] for open systems. In this sense, thepointwise geometric conditions in Remark 1 revisit and extendthe comparison between cooperative systems, incrementallypositive systems of [3, Section VIII], and the Kamke conditionprovided in [43, Chapter 3].

4) One dimensional continuous-time systems are differen-tially positive: This property is well-known for systems in R:solutions are partially ordered because they cannot “pass eachother”. It remains true on closed manifolds such as S, eventhough the conal order does not induce a (globally defined)partial order in that case.

5) Non-constant cones for oscillating dynamics: Movingfrom constant to non-constant cone fields opens the way tothe analysis of more general limit sets such a oscillationsor limit cycles. The harmonic oscillator studied in Section IIprovides a first simple example of differential positivity withrespect to a non-constant cone field. In particular, considerKX (x) := {δx ∈ R2 \ {0} | k1(x, δx) ≥ 0, k2(x, δx) ≥ 0},where k1(x, δx) := −(x1 + x2)δx1 + (x1 − x2)δx2 andk2(x, δx) := −(x2 − x1)δx1 + (x1 + x2)δx2. The cone fieldis well defined on the (invariant) manifold X := R2 \ {0}.Differential positivity with respect to KX (x) follows from thegeometric conditions in Remark 1, since k1 = 0 and k2 = 0everywhere.

6

The differential positivity of the harmonic oscillator withrespect to KX (x) is not surprising if one looks at the rep-resentation of the oscillator in polar coordinates ϑ = 1,ρ = 0. The state manifold becomes the cylinder S × R+

and the system decompose into two one-dimensional systems,which suggests the invariance of any cone field rotating withϑ, as shown in Figure 5 (left). Indeed, polar coordinatessuggest differential positivity for arbitrary decoupled dynamicsϑ = f(ϑ), ρ = g(ρ) with respect to the cone field K(ϑ, ρ) :={(δϑ, δρ) ∈ R2) | δϑ ≥ 0 , δρ ≥ 0}. In fact, the linearizationreads ˙δϑ = ∂f(ϑ)δϑ, δρ = ∂g(ρ)δρ, which guarantees that˙δϑ = 0 for δϑ = 0 and δρ = 0 for δρ = 0, as required by

Remark 1.Possibly, the invariance of the cone field can be strengthened

to contraction by combining the two uncoupled dynamics. Forexample, when f(ϑ) = 1 and g(ρ) = ρ− ρ3

3 , the trajectories ofthe variational dynamics move towards the interior of the conefield K(ϑ, ρ) := {(δϑ, δρ) ∈ R2) | δϑ ≥ 0 , δϑ2 − δρ2

ρ2 ≥ 0}.In fact, d

dt (δϑ2− δρ2

ρ2 ) = (1 + ρ2

3 ) δρ2

ρ2 > 0 for each (δϑ, δρ) ∈bdK(ϑ, ρ) \ {0}. Figure 5 (right) provides a representation ofthe (projective) contraction of the cone. We anticipate that thiscontraction property is tightly connected to the existence of aglobally attractive limit cycle.

-2 0 2

-2

-1

0

1

2

3

x2(t

)

x1(t)

-2 0 2

-2

-1

0

1

2

3

x2(t

)

x1(t)

Figure 5. Simulations of the variational dynamics δΣ of the two cases(i) ρ = 0 (left) and (ii) ρ = (ρ− ρ3

3) (right), from the same initial condition.

The red arrows represent the motion of the variational component along thetrajectory.

Remark 2: Differential positivity requires classical posi-tivity of the linearized dynamics at fixed points. In fact,Definition 1 shows that the cone field at any fixed pointx∗ is given by the invariant cone of the (positive) linearizeddynamics at x∗. The harmonic oscillator is not a positive linearsystem, because of the presence of the two complex eigen-values. Thus, it is not differentialy positive in R2. However,polar coordinates reveal that it is differentially positive in themanifold X = R2 \ {0}. y

VI. DIFFERENTIAL PERRON-FROBENIUS THEORY

A. Contraction of the Hilbert metric

Bushell [11] (after Birkhoff [8]) used the Hilbert metricon cones to show that the strict positivity of a mappingguarantees contraction among the rays of the cone, openingthe way to many contraction-based results in the literature ofpositive operators [11], [34], [42], [9], [29], among whichthe reduction of the Perron-Frobenius theorem to a specialcase of the contraction mapping theorem [27], [11], [8].

Taking inspiration from these important results, we rely onthe infinitesimal contraction properties of the Hilbert metricto study the contraction properties of differentially positivesystems.

Consider the product manifold X ×U where X is endowedwith the cone field KX (x, u) ⊆ TxX , for each (x, u) ∈ X ×U . Following [11], for any given (x, u), take any δx, δy ∈KX (x, u) \ {0} and define the quantities

MKX (x,u)(δx, δy) := inf{λ ∈ R≥0 |λδy − δx ∈ KX (x, u)}mKX (x,u)(δx, δy) := sup{λ ∈ R≥0 | δx− λδy ∈ KX (x, u)}.

(11)MKX (x,u)(δx, δy) := ∞ when {λ ∈ R≥0 |λδy − δx ∈KX (x, u)} = ∅. The Hilbert’s metric dKX (x,u) induced byKX (x, u) is given by

dKX (x,u)(δx, δy) = log

(MKX (x,u)(δx, δy)

mKX (x,u)(δx, δy)

). (12)

In each cone KX (x, u) dKX (x,u) is a projective distance:for each δx, δy, δz ∈ KX (x, u), dKX (x,u)(δx, δy) ≥ 0,dKX (x,u)(δx, δy) = dKX (x,u)(δy, δx), dKX (x,u)(δx, δy) ≤dKX (x,u)(δx, δz) + dKX (x,u)(δz, δy), and dKX (x,u)(δx, δy) =0 if and only if δx = λδy with λ ≥ 0.

The following theorem is a generalization of Birkhoffresult: it shows that strict differential positivity guarantees theexponential contraction of the metric when the input u(·) actsuniformly on the system (a feedforward signal). The uniformaction of the input is modeled by taking the variational inputδu(·) = 0, since δu represents the infinitesimal mismatchbetween two inputs.

For readability, in what follows we denote the Hilbert metricalong a solution pair dKX (x(t),u(t))(·, ·) with d∗(t)(·, ·).

Theorem 2: Let Σ be a dynamical system on the state/inputmanifold X×U , differentially positive with respect to the conefield K(x, u) := KX (x, u)×{0}, where KX (x, u) ⊆ TxX foreach (x, u) ∈ X × U . Then, for all t ≥ t0,

d∗(t)(δx1(t), δx2(t)) ≤ d∗(t0)(δx1(t0), δx2(t0)) (13)

for any (x(·), δx1(·), u(·), 0), (x(·), δx2(·), u(·), 0) ∈ δΣ withdomain [t0,∞) and δx1(t0), δx2(t0) ∈ KX (x(t0), u(t0)).

If Σ is strictly differentially positive then there exist ρ ≥ 1and λ > 0 such that, for all t ≥ t0,

d∗(t)(δx1(t), δx2(t)) ≤ ρe−λ(t−t0)d∗(t0)(δx1(t0), δx2(t0))(14)

for any (x(·), δx1(·), u(·), 0), (x(·), δx2(·), u(·), 0) ∈ δΣ withdomain [t0,∞) and δx1(t0), δx2(t0) ∈ KX (x(t0), u(t0)).Moreover, d∗(t)(δx1(t), δx2(t)) <∞ for t ≥ t0 + T . y

B. The Perron-Frobenius vector field

The Perron-Frobenius vector of a strictly positive linear mapis a fixed point of the projective space. Its existence is aconsequence of the contraction of the Hilbert metric, [11]. Toexploit the generalized contraction of Theorem 2, we assumethat the input acts uniformly on the system, that is, δu(·) = 0.We endow the state manifold X with a (smooth) Riemannianstructure and we define B(x) := {δx ∈ TxX | |δx|x = 1} ⊆TxX , to make the following assumption.

7

Assumption 1: (KX (x, u) ∩ B(x), dKX (x,u)) is a completemetric space for any given (x, u) ∈ X × U 1.

To introduce the Perron-Frobenius vector field we studythe asymptotic behavior of δΣ, looking at solutions pairs(z(·), u(·)) ∈ Σ with domain I := (−∞, t) (backwardcompleteness of Σ). Recall that for any (z(·), u(·)) ∈ Σ,if δu(·) = 0 then δz(t) = ∂z(t0)ψ(t, t0, z(t0), u(·))δz(t0)is a trajectory of the variational components of δΣ along(z(·), u(·)).

Theorem 3: Let Σ be a dynamical system on the state/inputmanifold X ×U . Suppose that Σ is strictly differentially posi-tive with respect to the cone field K(x, u) := KX (x, u)×{0}such that KX (x, u) ⊆ TxX for each (x, u) ∈ X × U andsuppose that Assumption 1 holds.

For any input u(·) : R → U which makes Σ backwardcomplete, there exists a time-varying vector field wu(·)(x, t) ∈intKX (x, u(t)) ∩ B(x), x ∈ X and t ∈ R, such that anysolution pair (z(·), u(·)) ∈ Σ satisfies

limt0→−∞

∂z(t0)ψ(t, t0, z(t0), u(·))KX (z(t0), u(t0)) =

= {λwu(·)(z(t), t) |λ ≥ 0} .(15)

We call this vector field the Perron-Frobenious vector field. yCorollary 1: Under the assumptions of Theorem 3, if

u(·) = u ∈ U (constant), then the Perron-Frobenius vectorfield reduces to a continuous time-invariant vector field wu(x).For linear systems the Perron-Frobenius vector field reducesto the (constant) Perron-Frobenius vector. y

The evolution of an initial cone KX (z(t0), u(t0)) along the(variational) flow of the system asymptotically converges tothe span of the Perron-Frobenius vector field wu(·)(x, t) at-tached to each x ∈ X , as illustrated in Figure 6. Decomposingthe variational trajectory δz(·) along z(·) into a directionalcomponent ϑ(t) := δz(t)

|δz(t)|z(t) ∈ KX (z(t), u(t)) ∩ Bz(t), anda magnitude component ρ(t) := |δz(t)|z(t), Theorem 3 estab-lishes that ϑ(t) is guaranteed to converge to wu(·)(z(t), t),for any initial condition ϑ(t0).

For constant inputs the Perron-Frobenius vector fieldhas a simple geometric characterization. Take any tra-jectory (x(·), δx(·)) of the prolonged system δΣ underthe action of the constant input u, and suppose thatd(x(t),u)(wu(x(t)), δx(t)) = 0 for some t ∈ R. Then,from (14), d(x(t+τ),u)(wu(x(t + τ)), δx(t + τ)) = 0 foreach τ ≥ 0, which shows that wu(x(t)) must be a time-reparametrized trajectory of Σ. Therefore, wu(x) belongs tointKX (x, u) for all x and satisfies the partial differential equa-tion [∂xwu(x)]f(x, u) = [∂xf(x, u)]wu(x) − λ(x, u)wu(x)for continuous time systems, for some λ(x, u) ∈ R whichguarantees |wu(x)|x = 1. In a similar way, for discretedynamics we have wu(f(x, u)) = λ(x, u)[∂xf(x, u)]w(x, u).As before, λ(x, u) ∈ R is selected to guarantee |wu(x)|x = 1.Existence and uniqueness of the solution wu(x) follow fromthe contraction of the Hilbert metric, under the assumption ofbackward and forward invariance of X .

1 The reader is referred to [11, Section 4], [29, Section 2.5], or [49] forexamples of complete metric spaces on cones.

z(t)

KX (z(t0), u(t0))z(t0)

z(t′0)

wu(·)(t, x)

KX (z(t′0), u(t′0))

KX (z(t), u(t))

z(·)

Figure 6. The contraction at time t from different initial cones, fort0 < t′0 < t. Note that ∂z(t0)ψ(t, t0, z(t0), u(·))KX (z(t0), u(t0)) ⊆∂z(t′0)

ψ(t, t′0, z(t′0), u(·))KX (z(t′0), u(t′0)) ⊆ KX (z(t), u(t)). At time t,

for t− t0 →∞, the cone reduces to a line.

VII. LIMIT SETS OF (CLOSED)DIFFERENTIALLY POSITIVE SYSTEMS

A. Behavior dichotomy

For closed continuous-time dynamical systems (or opencontinuous-time systems with constant inputs) the combinationof the local order on the system state manifold and theprojective contraction of the variational dynamics toward thePerron-Frobenius vector field w(x) restrict the asymptoticbehavior of differentially positive systems. The next theoremcharacterizes the ω-limit sets of those systems.

Theorem 4: Let Σ be a closed continuous (complete) sys-tem x = f(x) with state manifold X , strictly differentiallypositive with respect to the cone field KX (x) ⊆ TxX . UnderAssumption 1, suppose that the trajectories of Σ are bounded.Then, for every ξ ∈ X , the ω-limit set ω(ξ) satisfies one ofthe following two properties:(i) The vector field f(x) is aligned with the Perron-

Frobenius vector field w(x) for each x ∈ ω(ξ) (i.e.f(x) = λ(x)w(x), λ(x) ∈ R), and ω(ξ) is either afixed point or a limit cycle or a set of fixed points andconnecting arcs;

(ii) The vector field f(x) is not aligned with the Perron-Frobenius vector field w(x) for each x ∈ ω(ξ) such thatf(x) 6= 0, and either lim inf

t→∞|∂xψ(t, 0, x)w(x)|ψ(t,0,x) =

∞ or limt→∞

f(ψ(t, 0, x)) = 0. yThe interpretation of Theorem 4 is that the asymptotic

behavior of Σ is either described by a Perron-Frobeniuscurve γw(·), that is, a curve γw(s) = w(γw(s)) for alls ∈ dom γw(·); or is the union of the limit points ofsome trajectory ψ(·, 0, ξ), ξ ∈ X , nowhere tangent to thePerron-Frobenius vector field, as clarified in Section VII-C,and characterized by high sensitivity with respect to initialconditions, because of the unbounded linearization. The proofof Theorem 4 in Appendix, Section B, is of interest on its ownsince it illustrates how differential Perron-Frobenius theoryimpacts the behavior of Σ. In the next two subsections wefurther discuss the implications of Theorem 4 in case (i) andin case (ii), respectively.

B. Simple attractors of differentially positive systems

A first consequence of Theorem 4 is a result akin toPoincare-Bendixson characterization of limit sets of planarsystems.

8

Corollary 2: Under the assumptions of Theorem 4, con-sider an open, forward invariant region C ⊆ X that does notcontain any fixed point. If the vector field f(x) ∈ intKX (x)for any x ∈ C, then there exists a unique attractive periodicorbit contained in C. y

The result shows the potential of differential positivity forthe analysis of limit cycles in possibly high dimensionalspaces. Since stable limit cycles must correspond to Perron-Frobenius curves, stable limit cycles are excluded whenPerron-Frobenius curves are open, a property always satisfiedin vector spaces with constant cone field. For a differentiallypositive system defined in a vector space, the cone field mustnecessarily “rotate” with the periodic orbit in order to allowfor limit cycle attractors (see, for example, Section V-B5).

Beyond isolated fixed point and limit cycles, the limit setsof differentially positive systems are severely restricted by(local) order properties, see Figure 7 for an illustration. Inparticular, the intuitive argument ruling out homoclinic orbitslike in Figure 2 is made rigorous with Theorem 4. A limitset given by a connecting arc between two hyperbolic fixedpoints can exists only if it is everywhere tangent to the Perron-Frobenius vector field (Theorem 4.i), or nowhere tangent tothe Perron-Frobenius vector field (Theorem 4.ii). Because anyorbit between two hyperbolic fixed points must belong tothe unstable manifold of its α-limit set and to the stablemanifold of its ω-limit set, it can be a Perron-Frobeniuscurve only if, whenever it is tangent to the Perron-Frobeniuseigenvector of its α-limit, it is also tangent to the Perron-Frobenius eigenvector of its ω-limit.

Corollary 3: Under the assumptions of Theorem 4, con-sider an orbit that connects two hyperbolic fixed points ye, ze,respectively as t→ −∞ and t→∞. If the orbit is tangent tow(ye) at ye, then it is tangent to w(ze) at ze. y

The corollary rules out the possibility of a homoclinic orbitwith a one-dimensional unstable manifold, a typical ingredientof strange attractors. For system depending on parameters, thecorollary rules out the possibility of homoclinic bifurcations[46, Chapter 8] where the homoclinic orbit is tangent to thedominant eigenvector of the saddle point. In accordance withTheorem 4, a limit set given by a homoclinic orbit can onlyexist if it is nowhere tangent to the Perron-Frobenius vectorfield, which rules out the possibility of being part of a simpleattractor. The two situations are illustrated in Fig 8.

C. Complex limit sets of differentially positive systems are notattractors.

Part (ii) of Theorem 4 allows for more complex limit setsthan those described in Part (i), but those limit sets cannot beattractors, because they are nowhere tangent to the dominantdirection of the linearization. This property has been wellstudied for monotone systems. For instance, Smale proposeda construction to imbed chaotic behaviors in a cooperativeirreducible system [43, Chapter 4]. The transversality of thoselimit sets to the Perron-Frobenius vector field extends to thetrajectories that converge to them. For instance, consider anyω-limit set ω(ξ), ξ ∈ X , satisfying Part (ii) of Theorem 4.Any trajectory whose ω-limit points belong to ω(ξ) is nowhere

Figure 7. Simple attractors consistent with the ordering property of a Perron-Frobenius curve. Cones are shaded in blue. Top-left: fixed point with dominanteigenvector. Top-right: bistable systems, the heteroclinic orbits connectingthe saddle (red) to the stable fixed points (blue) coincides with the imageof some Perron-Frobenius curves. Bottom: limit cycles (left) or fixed pointsand connecting arcs (right) coincides with the image of some closed Perron-Frobenius curve, thus require require a rotating cone field (on vector spaces).

Figure 8. The stable and unstable manifolds of the saddle have dimension 1and 2, respectively. The Perron-Frobinius vector field is represented in red. Thelimit set given by the homoclinic orbit on the right part of the figure (dashedline) is ruled out by Corollary 3. The limit set given by the homoclinic orbiton the left part of the figure (solid line) is compatible with Theorem 4 butthe Perron-Frobenius vector field is nowhere tangent to the curve.

tangent to the Perron-Frobenius vector field. Moreover, if thetrajectory does not converge to a fixed point then it shows highsensitivity with respect to initial conditions.

Corollary 4: Under the assumptions of Theorem 4, supposethat for some ξ ∈ X , ω(ξ) satisfies Part (ii) of Theorem4. Then, for any z ∈ X such that ω(z) ⊆ ω(ξ), thetrajectory ψ(·, 0, z) satisfies f(ψ(t, 0, z)) /∈ KX (ψ(t, 0, z)) \{0} for each t ≥ 0. If ω(z) is not a singleton, thenlim inft→∞

|∂zψ(t, 0, z)w(z)|ψ(t,0,z) =∞. y

The reason why the possibly complex limit sets of differen-tially positive systems are of little importance for the overallbehavior is that their basin of attraction W seems stronglyrepelling. In accordance to Corollary 4, it is very “likely” fora trajectory in a small neighborhood ofW to move away fromW along the Perron-Frobenius vector field and “unlikely” toreturn to W at later time. The argument can be made rigorousfor strongly order preserving monotone systems, allowing torecover the following celebrated result for monotone systems[43], [23].

Corollary 5: Let Σ be a continuous dynamical system ofthe form x = f(x) on a vector space X , strict differentiallypositive with respect to the constant cone field KX ⊆ TxX =

9

X . Under boundedness of trajectories, the ω-limit set ω(ξ) isa fixed point for almost all ξ ∈ X . yFor general differentially positive systems, the above discus-sion leads to the following conjecture.

Conjecture 1: Under the assumptions of Theorem 4, foralmost every ξ ∈ X , the ω-limit set ω(ξ) is given by eithera fixed point, or a limit cycle, or fixed points and connectingarcs. y

The implication of Conjecture 1 would be that any limit setnot covered by case (i) in Theorem 4 could at best attract aset of initial conditions of zero measure.

VIII. EXTENDED EXAMPLE: DIFFERENTIAL POSITIVITY OFTHE DAMPED PENDULUM

The results of the paper are briefly illustrated on the analysisof the classical (adimensional) nonlinear pendulum model:

Σ :

{ϑ = vv = − sin(ϑ)− kv + u

(ϑ, v) ∈ X := S×R ,

(16)where k ≥ 0 is the damping coefficient and u is the (constant)torque input.

The analysis of the state matrix A(ϑ, k) for ϑ ∈ S of thevariational system[

˙δϑ

δv

]=

[0 1

− cos(ϑ) −k

]

︸ ︷︷ ︸=:A(ϑ,k)

[δϑδv

](δϑ, δv) ∈ T(ϑ,v)X

(17)reveals that the pendulum is strictly differentially positive fork > 2 and differentially positive for k = 2 with respect to thecone field

KX (ϑ, v) := {(δϑ, δv) ∈ T(ϑ,v)X | δϑ ≥ 0, δϑ+ δv ≥ 0} .(18)

The differential positivity of (16) for k ≥ 2 has thefollowing simple geometric interpretation. For any k ≥ 2 andany value of ϑ, the matrix A(ϑ, k) has only real eigenvalues.The blue and the red lines in Figure 9 show the direction ofthe eigenvectors of A(ϑ, 4) (left) - A(ϑ, 3) (center) - A(ϑ, 2)(right), for sampled values of ϑ ∈ S. The blue eigenvectors(δϑ ≤ 0) are related to the smallest eigenvalues, which isnegative for each ϑ. The red eigenvectors (δϑ ≥ 0) arerelated to the largest eigenvalues. KX (ϑ, v) is represented bythe shaded area in Figure 9. The black arrows represent thevector field of the variational dynamics along the boundaryof the cone. By continuity and homogeneity of the vectorfield on the boundary of the cone, Σ is strictly differentiallypositive for each k > 2. It reduces to a differentially positivesystem in the limit of k = 2. The loss of contraction in sucha case has a simple geometric explanation: one of the twoeigenvectors of A(0, 2) belongs to the boundary of the coneand the eigenvalues of A(0, 2) are both in −1. The issues isclear for u = 0 at the equilibrium xe := (0, 0). In such a caseA(0, 2) gives the linearization of Σ at xe and the eigenvaluesin −1 makes the positivity of the linearized system non strictfor any selection of KX (xe).

For k > 2 the trajectories of the pendulum are bounded. hevelocity component converges in finite time to the set V :=

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

δ v

δ ϑ

Figure 9. KX (ϑ, v) is shaded in gray. The two eigenvector of A(ϑ, k) fora selection ϑ ∈ S are represented in red and blue. Red eigenvectors - largesteigenvalues. Blue eigenvectors - smallest eigenvalues. Left figure: k = 4.Center figure: k = 3. Right figure: k = 2.

{v ∈ R | − ρ |u|+1k ≤ v ≤ ρ |u|+1

k }, for any given ρ > 1,since the kinetic energy E := v2

2 satisfies E = −kv2 + v(u−sin(ϑ)) ≤ (|u| + 1 − k|v|)|v| < 0 for each |v| > |u|+1

k . Thecompactness of the set S×V opens the way to the use of theresults of Section VII. For u = 1 + ε, ε > 0, we have thatv ≥ ε − kv which, after a transient, guarantees that v > 0,thus eventually ϑ > 0. Denoting by f(ϑ, v) the right-handside in (16), it follows that, after a finite amount of time,every trajectory belongs to a forward invariant set C ⊆ S×Vsuch that f(ϑ, v) ∈ intKX (ϑ, v). By Corollary 2, there is aunique attractive limit cycle in C.

It is of interest to interpret differential positivity againstthe textbook analysis [46]. Following [46, Chapter 8], Figure10 summarizes the qualitative behavior of the pendulum fordifferent values of the damping coefficient k ≥ 0 and of theconstant torque input u ≥ 0 (the behavior of the pendulumfor u ≤ 0 is symmetric). The nonlinear pendulum cannotbe differentially positive for arbitrary values of the torquewhen k ≤ kc. This is because the region of bistable behaviors(coexistence of small and large oscillations) is delineated bya homoclinic orbit, which is ruled out by differental positivity(Corollary 3). For instance, looking at Figure 10, for anyk < kc there exists a value u = uc(k) for which the pendulumencounters a homoclinic bifurcation (see [46, Section 8.5] andFigures 8.5.7 - 8.5.8 therein). In contrast, the infinite-periodbifurcation at k > 2, u = 1 [46, Chapter 8] is compatible withdifferential positivity.

It is plausible that the “grey” area between kc and k = 2 is aregion where the nonlinear pendulum is differentially positiveover a uniform time-horizon rather than pointwise. A detailedanalysis of the region kc ≤ k < 2 is postponed to a furtherpublication.

u

k

fixed points

limit cycle

bistability1

kc

infinite-period

homoclinic

k = 2

Figure 10. The qualitative behavior of the pendulum for large/small valuesof torque and damping, as reported in [46, p. 272].

10

IX. CONCLUSIONS

The paper introduces the concept of differential positiv-ity, a local characterization of monotonicity through the in-finitesimal contraction properties of a cone field. The theoryof differential positivity reduces to the theory of monotonesystems when the state-space is linear and when the conefield is constant. The differential framework allows for ageneralization of the Perron-Frobenius theory on nonlinearspaces and/or non constant cone fields. The paper focuseson the characterization of limit sets of differentially positivesystems, showing that those systems enjoy properties akin tothe Poincare-Bendixson theory of planar systems. In particular,differential positivity is seen as a novel analysis tool forthe analysis of limit cycles and as a property that precludescomplex behaviors in a significant class of nonlinear systems.Many issues of interest remain to be addressed beyond thematerial of the present paper. The most pressing of thoseis probably the topic of feedback interconnections: negativefeedback interconnections of monotone systems are known toprovide a key mechanism of oscillation [21], [20] and it isappealing to analyze their differential positivity by inferringa (non-constant) cone field from the order properties of thesubsystems and from the interconnection structure only. Moregenerally, the construction of particular cone fields for inter-connections of relevance in system theory (e.g. Lure systems)as well as the relationship between differential positivity andhorizontal contraction recently studied in [19] will be the topicof further research.

APPENDIX

A. Proofs of Section VI

Proof of Theorem 2: Using Γ(x1,u1,x2,u2) to denote thelinear and invertible mapping Γ(x1, u1, x2, u2) that satis-fies Γ(x1, u1, x2, u2)KX (x1, u1) = KX (x2, u2) for each(x1, u1), (x2, u2) ∈ X × U (see Section IV), and d∗(x,u) todenote dKX (x,u) (for readability), note that the Hilbert metricsatisfies

d∗(x2,u2)(v, w) = d∗(x1,u1)(Γ−1(x1,u1,x2,u2)v,Γ

−1(x1,u1,x2,u2)w)

(19)for any (x1, u1), (x2, u2) ∈ X × U , and any v, w ∈KX (x2, u2). (19) follows by the combination of (11) withthe identity Γ−1

(x1,u1,x2,u2)bdKX (x2, u2) = bdKX (x1, u1) (bylinearity).

Along any given solution pair (x(·), u(·)) : [t0,∞)→ X ∈Σ define the linear operator

A(x(·),u(·))[τ2, τ1] :=

:= Γ−1(x(τ1),u(τ1),x(τ2),u(τ2))∂x(τ1)ψ(τ2, τ1, x(τ1), u(·))

(20)where t0 ≤ τ1 ≤ τ2. Thus, using d∗(t) to denote dKX (x(t),u(t)),and recalling that, in Theorem 2, x(t) = ψ(t, t0, x(t0), u(·)),δx1(t) = ∂x(t0)ψ(t, t0, x(t0), u(·))δx1(t0), and δx2(t) =∂x(t0)ψ(t, t0, x(t0), u(·))δx2(t0), for each t ≥ t0 we get

d∗(t)(δx1(t), δx2(t)) == d∗(t0)(A(x(·),u(·))[t, t0]δx1(t0), A(x(·),u(·))[t, t0]δx2(t0))≤ d∗(t0)(δx1(t0), δx2(t0)) .

(21)

The identity follows by the combination of (19), (20), anddifferential positivity. The inequality follows from the fact thatA(x(·),u(·))[t, t0] is a linear operator in KX (x(t0), u(t0)) →KX (x(t0), u(t0)), as in [27], [11].

Strict differential positivity guarantees that there exists T >0 such that, for any given τ ∈ [t0,∞),

A(x(·),u(·))[τ + T, τ ]KX (x(τ), u(τ)) ⊆ RX (x(τ), u(τ)) .(22)

Thus, following [11], [27], define projective diameter ∆T :=sup{dKX (x,u)(v1, v2) | v1, v2 ∈ RX (x, u)} <∞ and contrac-tion ratio µT := tanh

(∆T

4

)< 1. Then, using again d∗(t) to

denote dKX (x(t),u(t)), [11, Theorem 3.2] and [27, Proposition3.14] guarantee the inequality

d∗(τ+T )(δx1(τ + T ), δx2(τ + T )) == d∗(τ)(A(x(·),u(·))[τ+T,τ ]δx1(τ), A(x(·),u(·))[τ+T,τ ]δx2(τ))≤ µT d∗(τ)(δx1(τ), δx2(τ)) ,

(23)for all τ ≥ t0. By the semigroup property, for any integer k,and t ≥ t0 + kT we get

d∗(t)(δx(t), δy(t)) ≤ µkT d∗(t0)(δx(t0), δy(t0)) (24)

which establishes the exponential convergence.Finally, combining (24) and (22), for all t ≥ t0 + (k+ 1)T

we get d∗(t)(δx(t), δy(t)) ≤ µkT∆T . �

Proof of Theorem 3: Consider the solution pair (z(·), u(·)) ∈Σ such that z(t) = x and define C(t, t0, x, u(·)) :=∂z(t0)ψ(t, t0, z(t0), u(·))KX (z(t0), u(t0)). By construction,strict differential positivity guarantees that

C(t, t− T2, x, u(·)) ⊆ C(t, t− T1, x, u(·)) (25)

for each T2 ≥ T1 ≥ 0. Also, from Theorem 2 there existsT > 0 such that

lim supk→∞

{dKX (x,u(t))(v1,v2) | v1,v2∈C(t, t−kT, x, u(·))} = 0,

(26)since the right-hand side is bounded from above bylimk→∞ µk−1

T ∆T , where ∆T < ∞ and µT < 1 are respec-tively the projective diameter and the contraction ratio definedin the proof of Theorem 2.

Following [38, Chapter 4, Exercise 4.3], the combination of(25) and (26) guarantees that the set C(t, t−kT, x, u(·))∩B(x)converges to the singleton {wu(·)(x, t)} ⊆ intKX (x, u(t)) ∩B(x) as k → ∞. We write wu(·)(x, t) since the limit of theset C(t, t − kT, x, u(·)) ∩ B(x) for k → ∞ depends only onthe input signal u(·), the state x, and the time t. �

Proof of Corollary 1: To show time-invariance, consideru(·) = u. Under the action of the constant input u, con-sider the trajectories z(·) and y(·) such that z(t) = x andy(t + T ) = x, for some t, T ∈ R. Using B(x) :={δx ∈ TxX | |δx|x = 1} ⊆ TxX , from Theorem 3,

limt0→−∞

(∂z(t0)ψ(t, t0, z(t0),u)KX (z(t0),u))∩B(x)= wu(x, t)

and limt0→−∞

(∂y(t0)ψ(t+T, t0, y(t0), u)KX (y(t0), u))∩B(x) =

wu(x, t + T ). However, for constant u, Σ is time invariant,therefore uniqueness of trajectories from initial conditionsguarantees that z(τ) = y(τ + T ) for all τ ∈ (−∞, t), which

11

guarantees KX (z(t0), u)) = KX (y(t0 + T ), u)). wu(x, t) =wu(x, t+ T ) follows.

To show continuity, consider the family of vector fields {gk}given by g0(x) ∈ KX (x, u)∩B(x) and gk+1(ψ(T, 0, x, u)) :=∂xψ(T,0,x,u)gk(x)|∂xψ(T,0,x,u)gk(x)|x . Note that each gk is a continuousvector field because ∂xψ(T, 0, x, u) is continuous in xand the Riemannian structure is smooth in x. Moreover,dKX (x,u)(gk+1(x),wu(x)) ≤ µkT∆T for all x ∈ X and allk ≥ 1, where µT and ∆T are respectively the contraction ratioand the projective diameter defined in the proof of Theorem2. Indeed, gk converge uniformly to wu (with respect to theHilbert metric at x).

By contradiction, suppose that the Perron-Frobenius vectorfield is not continuous at x. Then, following [10, Definition2.1], in local coordinates, the ith component [wu(x)]i ofwu(x) is not a continuous function: there exist ε > 0, asequence of points yj → x as j → ∞, and a bound Jsuch that the absolute value |[wu(x)]i − [wu(yj)]i| ≥ ε forany j ≥ J . By the uniform convergence of gk to wu, thereexists a bound K such that |[gk(x)]i − [wu(x))]i| ≤ ε

3 and|[gk(yj)]i−[wu(yj))]i| ≤ ε

3 for all k ≥ K and all j. Therefore,|[gk(x)]i − [gk(yj)]i| ≥ ε

3 for all k ≥ K and j ≥ J , whichcontradicts the continuity of gk.

Finally, the coincidence between the Perron-Frobenius vec-tor field and the Perron-Frobenius vector for linear systems isa straightforward consequence of (15). �

B. Proofs of Section VII

For readability, in what follows we use ψt(x) := ψ(t, 0, x),∂ψt(x) := ∂xψ(t, 0, x), and dx(·, ·) := dKX (x)(·, ·). Recallthat the pair (ψt(x), ∂ψt(x)δx) is the trajectory of the pro-longed system δΣ given by x = f(x), ˙δx = ∂f(x)δx fromthe initial condition (x, δx) ∈ TX .

We develop first some technical results. The claims of thenext two lemmas are about the boundedness of the trajectoriesof the variational systems. The claims holds for both contin-uous and discrete systems (closed or with constant inputs).

Lemma 1: Let u(·) = u be constant. Under the assumptionsof Theorem 3, for any x ∈ X and any δx ∈ TxX , if|∂ψt(x)w(x)|ψt(x) <∞ then lim sup

t→∞|∂ψt(x)δx|ψt(x) <∞.

Lemma 2: Let u(·) = u be constant. Under the assumptionsof Theorem 3, let x be any point of X and suppose that thereexists δx ∈ intKX (x) such that lim sup

t→∞|∂ψt(x)δx|ψt(x) <∞.

Then, lim supt→∞

|∂ψt(x)w(x)|ψt(x) <∞.

Proof of Lemma 1: For the first item suppose that the impli-cations does not hold and |∂ψt(x)δx|ψt(x) grows unbounded.Take the vector δy = δx+αw(x). Note that for α sufficientlylarge δy ∈ KX (x). These facts and the linearity of ∂ψt(x)guarantee that |∂ψt(x)δy|ψt(x) grows unbounded and thereexists t sufficiently large such that either (i) ∂ψt(x)δy /∈KX (ψt(x)), which contradicts differential positivity, or (ii)∂ψt(x)δy ' ρψt(x)w(x) where β ∈ R is a scaling fac-tor. Thus, for all t ≥ t, by linearity, |∂ψt(x)δy|ψt(x) 'ρ|ψt(x)w(x)|ψt(x) which grows unbounded contradicting theassumption on |ψt(x)w(x)|ψt(x). �

Proof of Lemma 2: For the second item, consider any de-composition δx = αw(x) + βδz where α, β ∈ R≥0 andδz ∈ KX (x), which can always be achieved for α suf-ficiently small since δx ∈ intKX (x). Then, ∂ψt(x)δx =∂ψt(x)[αw(x) + βδz] = α∂ψt(x)w(x) + β∂ψt(x)δz and,by projective contraction, ∂ψt(x)δz converges asymptoticallyto ρtw(ψt(x)) for some ρt ∈ R≥0. Thus,

lim supt→∞

|∂ψt(x)δx|ψt(x) =

= lim supt→∞

|α∂ψt(x)w(x) + βρtw(ψt(x))|ψt(x)

= lim supt→∞

∣∣∣α∂ψt(x)w(x) + βρt∂ψt(x)w(x)

|∂ψt(x)w(x)|ψt(x)

∣∣∣ψt(x)

= lim supt→∞

(α+ βρt

|∂ψt(x)w(x)|ψt(x)

)|∂ψt(x)w(x)|ψt(x)

≥ α lim supt→∞

|∂ψt(x)w(x)|ψt(x) . (27)�

The next lemma shows that any trajectory of a continuousand closed differentially positive system whose motion followsthe Perron-Frobenius vector field either converges to a fixedpoint or defines a periodic orbit. In what follows we will useψt(x) := ψ(t, 0, x) and ∂ψt(x) := ∂xψ(t, 0, x).

Lemma 3: Under the assumptions of Theorem 4, considerany x such that, for all t, f(ψt(x)) = λ(ψt(x))w(ψt(x)) and|λ(ψt(x))| ≥ ρ > 0. Then, the trajectory ψt(x) is periodic.Proof of Lemma 3: In what follows we use A := {ψt(x) | t ∈R} and Bε(x) to denote a ball of radius ε centered at x: for anytwo points z in Bε(x) there exists a curve γ(·) such that γ(0) =

x, γ(1) = z and whose length L(γ(·)) =∫ 1

0|γ(s)|γ(s)ds ≤ ε.

We make also use of the notion of local section at x, which isany open set S ⊆ X of dimension n−1 (X has dimension n)contained within a (sufficiently) small neighborhood Bε(x) ofx such that x ∈ S and w(z) /∈ TzS for each z ∈ S. Finally,for any given Rimannian tensor such that 〈δx,w(x)〉x ≥ 0for any x ∈ X and δx ∈ KX (x), define the vertical projectionWx(δx) := 〈δx,w(x)〉xw(x), and the horizontal projectionHx(δx) := δx−Wx(δx).

1) Bounded variational dynamics: 0 6= f(ψt(x)) =λ(ψt(x))w(ψt(x)) for all t ≥ 0 therefore, by continuity ofthe vector field and boundedness of trajectories, for ε > 0sufficiently small, f(z) ∈ intKX (z) or −f(z) ∈ intKX (z)for all z ∈ E :=

⋃t≥0 Bε(ψt(x)). Note that E is a compact

set by boundedness of trajectories. Without loss of generalityconsider f(z) ∈ intKX (z). Then, f(ψt(z)) ∈ intK(ψt(z))for all z ∈ E and t ≥ 0, by differential positivity combinedwith the identity f(ψt(z)) = ∂ψt(z)f(z), which makes(ψt(z), f(ψt(z))) a trajectory of the prolonged system.

By boundedness of trajectories, |∂ψt(z)f(z)|ψt(z) =|f(ψt(z))|ψtz is necessarily bounded. Lemma 2 guaranteesthat lim sup

t→∞|∂ψt(z)w(z)|ψt(z)<∞ for all z ∈ E . By Lemma

1, lim supt→∞

|∂ψt(z)δz|ψt(z)<∞ for all z ∈ E and δz ∈ TzX .

Similar results can be obtained for the case −f(z) ∈ intKX (z)exploiting the linearity of ∂ψt(z). Finally, consider any z ∈ Eand define αz := sup

δz∈TzX ,|δz|z=1

lim supt→∞

|∂ψt(z)δz|ψt(z) <∞.

Then, for any δz ∈ TzX , lim supt→∞

|∂ψt(z)δz|ψt(z) < αz|δz|z .

Since E is a compact set, there exists α := supz∈E αz .2) Contraction of the horizontal component: Take z ∈ E .

12

For δz ∈ KX (z), combining the contraction propertylimt→∞ dψt(z)(∂ψt(z)δz,w(ψt(z))) = 0 of Theorem 2and the bound lim sup

t→∞|∂ψt(z)δz|ψt(z) ≤ α|δz|z in 1),

we get limt→∞

∂ψt(z)δz − Wψt(z)(∂ψt(z)δz) = 0, that is,limt→∞

Hψt(z)(∂ψt(z)δz) = 0. A similar result holds forδz ∈ −KX (z). Consider now δz /∈ KX (z). Define thenew vector δz∗ = δz + αw(z). For α sufficiently largeδz∗ ∈ KX (z). Therefore lim

t→∞Hψt(z)(∂ψt(z)δz

∗) = 0 whichimplies lim

t→∞Hψt(z)(∂ψt(z)δz) = 0.

3) Attractiveness of ψt(x): Consider the case t = 0 sinceψ0(x) = x (the argument is the same for t > 0) and take anycurve γ(·) : [0, 1] → E such that γ(0) = x and L(γ(·)) = ε,and consider the evolution of γ(·) along the flow of the system,that is, ψt(γ(s)) for s ∈ [0, 1]. We observe that d

dsψt(γ(s))=[∂ψt(γ(s))]γ(s). Thus, by 1), lim sup

t→∞| ddsψt(γ(s))|ψt(γ(s)) ≤

α|γ(s)|γ(s), which guarantees lim supt→∞

L(ψt(γ(·))) ≤ εα.

By 2), limt→∞

Hψt(γ(s))(ddsψt(γ(s)))|ψt(γ(s)) = 0 for all s ∈

[0, 1]. Thus, ddsψt(γ(s)) either converges to zero or aligns to

the Perron-Frobenius vector field. Precisely, three cases mayoccur:

• limt→∞

ddsψt(γ(s))) = 0,

• limt→∞

dψt(γ(s))(w(ψt(γ(s))), ddsψt(γ(s))) = 0,

• limt→∞

dψt(γ(s))(w(ψt(γ(s))),− ddsψt(γ(s))) = 0.

Furthrmore, limt→∞

dψt(γ(s))(w(ψt(γ(s))), f(ψt(γ(s)))) = 0

since f(ψt(γ(s))) ∈ KX (ψt(γ(s))). Thus, in the limit, theimage of ψt(γ(·)) is given by the image of a (time-dependentPerron-Frobenius) curve γwt (·) that satisfies either d

dsγwt (s) =

w(γwt (s)) or ddsγ

wt (s) = −w(γwt (s)) at any fixed t. By

construction, ddsγ

wt (s) = 1

λ(γwt (s))f(γwt (s)).

At each fixed t, γwt (·) is a (reparameterized) integral curveof the vector field f from the initial condition γwt (0) = ψt(x).Therefore, trajectory with initial condition in γ(s) convergesasymptotically to A, for all s ∈ [0, 1]. In particular, using thebound L(ψt(γ(·))) ≤ εα characterized above, each trajectoryconverges asymptotically to

Ct(x) :=

{ψt+τ (x) ∈ A | − αε

ρ≤ τ ≤ αε

ρ

}. (28)

4) Periodicity of the orbit: ψt(x) does not converge to afixed point and belongs to a compact set for each t, thereforethere exists a point ψt0(x) whose neighborhood Bε(ψt0(x))is visited by the trajectory infinitely many times for any givenε > 0. For simplicity, without any loss of generality, weconsider this point given at t0 = 0, that is, ψ0(x) = x.

Consider a local section S at x and consider the sequencetk → ∞ such that ψtk(x) ∈ S. Since f(x) is aligned withw(x), for ε sufficiently small, the continuity of the systemvector field f guarantees that S is transverse to f(z) for allz ∈ S ∩ Bε(x), that is, f(z) /∈ TzS ∩ Bε(x).

By 3), for every z ∈ S ∩ Bε(x), ψtk(z) converges asymp-totically to the set Ctk(x) as k →∞. For any positive integer

N , define the ( εN -inflated) set

C(ε/N)t (x) := {y | ∃γ : [0, 1]→ E ,∃s ∈ [0, 1] such that

γ(0) ∈ Ct(x), γ(s) = y, L(γ(·)) ≤ ε/N} .(29)

Then, by continuity, for every N > 0, there exists a k ≥ kNsufficiently large such that ψtk(z) ∈ C(ε/N)

tk(x) for all z ∈

S ∩ Bε(x).By the transversality of the section S with respect to the

system vector field, for N sufficiently large, we have thatthe flow from z ∈ C(ε/N)

tk(x) satisfies ψτ (z) ∈ S for some

−(αρ + δN )ε ≤ τ ≤ (αρ + δN )ε, where δN is some (small)positive constant such that δN → 0 as N →∞. Moreover, bycontinuity with respect to initial conditions, for N sufficientlylarge, we get

ψτ (z) ∈ S ∩ B ε3(ψtk(x)) . (30)

It follows that, for tk − (αρ + δN )ε ≤ t ≤ tk + (αρ + δN )ε, theflow ψt(·) maps every point of S∩Bε(x) into S∩B ε

3(ψtk(x)).

For k ≥ kN , denote by Pk the mapping from S ∩ Bε(x)into S ∩ B ε

3(ψtk(x)). Since ψtk(x) recursively visit any

local section of x, eventually, for some K ≥ kN , the flowsatisfies ψtK (x) ∈ S ∩ B ε

3(x). Using the results above, we

conclude that the flow of the system maps S ∩ Bε(x) intoS∩B ε

3(ψtK (x)) ⊆ S∩B 2ε

3(x), that is, PK is a contraction. By

Banach fixed-point theorem PK(x) = x, that is, ψtK (x) = x.�

We are now ready for the proof of the main theorem.Proof of Theorem 4: For any ξ ∈ X consider ω(ξ). Threecases may occur:1) f(x) = 0 for some x ∈ ω(ξ). x is a fixed point.2) f(x) ∈ intKX (x) \ {0} or −f(x) ∈ intKX (x) \ {0} forsome x ∈ ω(ξ). In such a case,

f(z) = λ(z)w(z) for all z ∈ ω(ξ) , (31)

where λ(z) ∈ R is a scaling factor. To see this, consider thecase f(x) ∈ intKX (x) (wlog). By definition of ω-limit set,there exists a sequence tk → ∞ such that lim

k→∞ψtk(ξ) = x.

For k ≥ k∗ sufficiently large, ψtk(ξ) belongs to an infinites-imal neighborhood of x therefore f(ψtk(ξ)) ∈ K(ψtk(ξ)) bycontinuity of the cone field since f(x) ∈ intKX (x). Then, byprojective contraction, lim

k→∞dψtk (ξ)(f(ψtk(ξ)), w(ψtk(x))) =

0, that is, f(x) = λ(x)w(x) for some scaling factor λ(x) ∈ R.By definition of ω-limit set, starting from tk∗ , it is possi-ble to find a sequence τk → ∞ as k → ∞ such thatlimk→∞

ψtk∗+τk(ξ) = z for any z ∈ ω(ξ). Thus, by the argumentabove, f(z) = λ(z)w(z) for all z ∈ ω(ξ). (31) guaranteesthat, for any x ∈ ω(ξ), the image of the trajectory ψt(x) isa subset of the image of some Perron-Frobenius curve γw(·).Note that λ(ψt(x)) may converge to zero. In such a case ψt(x)converges to a fixed point. Otherwise, |λ(ψt(x))| ≥ ε > 0therefore, by Lemma 3, ψt(x) is periodic.3) It remains to consider the case f(x) /∈ KX (x) (or−f(x) /∈ KX (x)) for some x ∈ ω(ξ). In such a case, fromthe previous item, f(z) /∈ KX (z) \ {0} for all z ∈ ω(ξ).Then, either limt→∞ f(ψt(x)) = 0 (ψt(x) converges to a

13

fixed point) or the contraction of the Hilbert metric enforceslim inft→∞

|∂ψt(x)w(x)|ψt(x) = ∞. For the latter, consider anysequence tk → ∞ as k → ∞ such that f(ψtk(x)) ≥ ε > 0.Take δx = f(x) + λw(x). For λ sufficiently large δx ∈KX (x). Then, lim

k→∞dψtk (x)(∂ψtk(x)δx, w(ψtk(x))) = 0 holds

only if the evolution of δx along the flow ∂ψtk(x)δx =∂ψtk(x)f(x)+λ∂ψtk(x)w(x) = f(ψtk(x))+λ∂ψtk(x)w(x)shows an unbounded growth of the component ∂ψtk(x)w(x).

�

Proof of Corollary 2: Recall that f(ψt(x)) = ∂ψt(x)f(x).Since f(x) ∈ intKX (x), Lemmas 1 and 2 guarantee thatlim supt→∞

|∂ψt(x)w(x)|ψt(x) < ∞. Since f(x) 6= 0 in C,

we conclude that Case (ii) of Theorem 4 does not occur.Exploiting again the assumption f(x) 6= 0 in C, Case (i) ofTheorem 4 guarantees that the trajectories of Σ converge toperiodic orbits. We need to prove the uniqueness.

By contradiction, suppose that A1 and A2 are two pe-riodic orbits such that A1 ∩ A2 = ∅. Take any curveγ(·) : [0, 1] → C such that γ(0) ∈ A1 and γ(1) ∈ A2,and recall that d

dsψt(γ(s)) = [∂ψt(x)x=γ(s)]γ(s). Sincef(ψt(x)) = ∂ψt(x)f(x) and f(x) ∈ intKX (x), Lemmas 1and 2 guarantee that lim sup

t→∞| ddsψt(γ(s))|ψt(γ(s)) <∞. Since

f(x) ∈ intKX (x) for any x ∈ C and the trajectories ofΣ are bounded, we can use the argument in 2) and 3) ofLemma 3 to show that d

dsψt(γ(s)) converges asymptotically toλa(ψt(γ(s)))w(ψt(γ(s))), thus to λb(ψt(γ(s)))f(ψt(γ(s))),for some (bounded) scaling factors λa(·), λb(·) ∈ R,

As a consequence, every trajectories whose initial condi-tions belongs to the image of γ(·) converges asymptoticallyto an integral curve of the system vector field f(x), forx ∈ C, connecting A1 and A2, since ψt(γ(0)) ∈ A1 andψt(γ(1)) ∈ A2 for all t ≥ 0. It follows that A1 ∩A2 6= ∅. Acontradiction. �

Proof of Corollary 3: For some x ∈ X , suppose that ye =limt→∞

ψ−t(x) and ze = limt→∞

ψt(x) are hyperbolic fixed point.Suppose that the orbit connecting ye to ze is tangential to

w(ye) at ye. Take now any point y in a small neighborhoodof ye such that y = ψ−T (x) for some T > 0. By continuity,f(y) ∈ intKX (y). Thus, lim

t→∞dψt(y)(f(ψt(y)),w(ψr(y))) =

limt→∞

dψt(y)(f(ψt(x)),w(xe)) = 0, by Theorem 2. �

Proof of Corollary 4: Consider the trajectory ψt(z). Followingthe proof of Corollary 3, necessarily, f(ψt(z)) /∈ KX (ψt(z))for any t ≥ 0. For instance, by contradiction, suppose thatf(ψt(z)) ∈ KX (ψt(z)) for some t ≥ 0. By definition,there exists a sequence tk → ∞ as k → ∞ such thatlimk→∞

ψt+tk(z) = x ∈ ω(ξ) thus ∂ψt+tk(z)f(z) = f(x) /∈KX (x). By continuity, since KX (x) is closed, there exists k∗

sufficiently large ∂ψt+tk∗ (z)f(ξ) /∈ KX (ψt+tk∗ (z)). But thiscontradicts differential positivity.

Suppose now that ω(z) ⊆ ω(ξ) is not a fixed point.Then, there exists a sequence tk → ∞ as k → ∞ suchthat limk→∞ f(ψtk(z)) = f(x) 6= 0 for some x ∈ ω(ξ).Take δz = f(z) + λw(z). For λ sufficiently large δz ∈KX (z). Then, lim

tk→∞dψtk (z)(∂ψtk(z)δz, w(ψtk(z))) = 0 holds

only if the evolution of δz along the flow ∂ψtk(z)δz =

∂ψtk(z)f(z)+λ∂ψtk(z)w(z) = f(x)+λ∂ψtk(z)w(z) showsan unbounded growth of the component ∂ψtk(z)w(z). �Proof of Corollary 5: Consider Part (i) of Theorem 4. Forany x ∈ ω(ξ), we have f(ψt(x)) = λ(ψt(x))w(ψt(x)).On vector spaces, for constant cone fields, closed curvescannot occur because every Perron-Frobenius curve is open.Therefore, lim

t→∞|λ(ψt(x))| = 0 by boundedness of solutions.

Consider Part (ii) of Theorem 4 and take any x ∈ ω(ξ). Eitherlimt→∞

f(ψt(x)) = 0, thus ψt(x) converges to a fixed point fort→∞, or lim inf

t→∞|∂ψt(x)w(x)|ψt(x) =∞.

This last case covers attractors which are not fixed points.We show that their basin of attraction has dimension n − 1at most. By contradiction, let A be an attractor with abasin of attraction BA of dimension n. By Corollary 4,from every x ∈ BA, lim inf

t→∞|∂ψt(x)w(x)|ψt(x) = ∞. Let

γw(·) be any Perron-Frobenius curve such that γw(0) =x. Since BA has dimension n, there exists an interval[s, s] 3 0 such that γw(s) ∈ BA for all s ∈ [s, s]. Also,lim inft→∞

|[∂ψt(x)x=γ(s)]γw(s)|ψt(x) =∞ for all s ∈ [s, s], that

is, lim inft→∞

L(ψt(γw(·))) =∞.

For each t > 0, the curve γt(·) := ψt(γw(·)) is a reparam-

eterization of a Perron-Frobenius curve, that is, ddsγt(s) =

λt(s)w(γt(s)) where λt(s) is a scalar. Thus, γt(·) is anopen curve for each t that grows unbounded as t → ∞. Itfollow that, for all s ∈ [s, s], the trajectory ψt(γw(s)) growsunbounded, contradicting the assumption on the boundednessof the trajectories of Σ. �

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